This is wrong. Using a magic item is it's own action, so casting a spell and using a magic item will allow you to technically cast 2 spells in the same round.
If you use a magic item that allows you to cast a spell, you're still casting a spell.
The action of using a magic item is not the same action as casting a spell. They are 2 distinct and different actions. The magic item is casting the spell NOT YOU. It's the same as using a potion.
Others have already provided the relevant quotes. Again, a wand typically
allows the user to cast a spell. It does not cast the spell itself. Potions, on the other hand, provide you with the benefits of a spell without allowing you to cast it.
...Of course some DMs may think otherwise, but that's just as dumb as the people who think a wizard can counter a counterspell.
You can. Nothing dumb about it (and please refrain from insulting those who play differently than you). Counterspell is a spell; you can counterspell any spell you see being cast. There's no "except those with a casting time of 1 reaction" exception.
I also feel you can't counter spell like abilities, since they also don't require components. They just happen.
5e does not have spell-like abilities. It has innate spellcasting, which
often allows you to cast without some components, but not always. For example, a dao's innate spellcasting allows it to cast certain spells without material components, but it still requires verbal and somatic components. Either way, though, the dao is still explicitly casting a spell.
I rule that magic initiate does provide a spell slot of the appropriate type. Like any spell slot it can be used for other things. And a Wizard Initiate can learn new 1st level spells to use the slot for.
If it’s not using a spell slot it doesn’t count towards the limit.
None of that is supported by the actual rules.
The restriction is for your entire round.
No, it's just for the turn you cast on. So if you cast a bonus action casting time spell on your turn, you can use a reaction casting time spell on the next turn.
Counterspelling for example, can only be done if you didn’t use a slot on your turn (or you can cast it without using a slot).
You cannot use a counterspell on your turn if you cast a spell with a bonus action casting time on your turn. Later in the round, you absolutely can. Slots have absolutely nothing to do with it. The rules limiting other spells if you cast a bonus action casting time spell always refer to casting a spell, not to expending a spell slot.
Correct on Paladin and Cleric, one spell slot using smite per turn.
This is also unsupported by the rules. All that said...
Note these are all my rulings and not intended to reflect RAW or RAI. I’ve also run games where I throw out the restriction all together.
Well, okay then!

But this significantly reduces the value of your post vis-a-vis how the rules actually work. Regardless, play however you like and however works for you and your table. But you might want to lead with this point, rather than tacking it onto the end of your post, so as not to lead people who are looking for how the rules work as written astray.